Infinite springs with force constant $k$, $2k$, $4k$ and $8k$.... respectively are connected in series. The effective force constant of the spring will be
A$2K$
B$k$
C$\frac{k}{2}$
D$\frac{k}{4}$
Medium
Download our app for free and get started
C$\frac{k}{2}$
c (c)$\frac{1}{{{k_{eff}}}} = \frac{1}{k} + \frac{1}{{2\,k}} + \frac{1}{{4\,k}} + \frac{1}{{8\,k}} + ....$
$ = \frac{1}{k}\left[ {1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + .....} \right]$$ = \frac{1}{k}\left( {\frac{1}{{1 - 1/2}}} \right)$$ = \frac{2}{k}$
(By using sum of infinite geometrical progression $a + \frac{a}{r} + \frac{a}{{{r^2}}} + ...\infty $ sum (S) $ = \frac{a}{{1 - r}}$)
$\therefore {k_{eff}} = \frac{k}{2}.$
Download our app
and get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Figure shows the circular motion of a particle. The radius of the circle, the period, sense of revolution and the initial position are indicated in the figure. The simple harmonic motion of the $x-$ projection of the radius vector of the rotating particle $P$ is
Speed $v$ of a particle moving along a straight line, when it is at a distance $x$ from a fixed point on the line is given by $v^2 = 108 - 9x^2$ (all quantities in $S.I.$ unit). Then
A particle executes $SHM$ of period $1.2\, sec$ and amplitude $8\, cm.$ Find the time it takes to travel $3\,cm$ from the positive extremity of its oscillation. ..... $\sec$
A circular disc of mass $10 \;kg$ is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be $1.5 \;s$. The radius of the disc is $15\; cm .$ Determine the torsional spring constant of the wire in $N\;m\;rad^{-1}$. (Torsional spring constant $\alpha$ is defined by the relation $J=-\alpha \theta,$ where $J$ is the restoring couple and $\theta$ the angle of twist).
A book is resting on a shelf that is undergoing vertical simple harmonic oscillations with an amplitude of $2.5 \,cm$. What is the minimum frequency of oscillation of the shelf for .......... $Hz$ the book will lose contact with the shelf? (Assume that, $g=10 \,ms ^{-2}$ )
A mass $\mathrm{m}$ is suspended from a spring of negligible mass and the system oscillates with a frequency $f_1$. The frequency of oscillations if a mass $9 \mathrm{~m}$ is suspended from the same spring is $f_2$. The value of $\frac{f_1}{f_{.2}}$ is_____________.
A particle executes $S.H.M.$ according to equation $x=10( cm ) \cos \left[2 \pi t+\frac{\pi}{2}\right]$, where $t$ is in second. The magnitude of the velocity of the particle at $t=\frac{1}{6} \,s$ will be .............. $cm / s$